MTH111/MAT111 Lecture 1: Elementary Set Theory PDF

Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment MTH111/MAT111- Lecture 1: Elementary Set Theory Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Overview Introduction to Set Theory...

Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment MTH111/MAT111- Lecture 1: Elementary Set Theory Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Overview Introduction to Set Theory Basic Terminology and Notation Types of Sets Set Operations Venn Diagrams Applications in Mathematics Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment What is a Set? Definition: A set is a well-defined collection of distinct objects, called elements. Sets are usually denoted by capital letters such as A, B, C , etc. Elements of a set are denoted by small letters or numbers and enclosed within curly braces {}. Examples: A = {1, 2, 3} B = {apple, banana, coconut, mango} Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Set Membership Set Membership Symbol: x ∈ A: Means that x is an element of set A. x∈ / A: Means that x is not an element of set A. Example: If A = {1, 2, 3}, then: 2∈A 4∈ /A Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example - Solution Question: Given the set A = {2, 4, 6, 8}, check if the following statements are true or false: 1 4∈A 2 5∈A 3 9∈ /A Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example Question: Given the set A = {2, 4, 6, 8}, check if the following statements are true or false: 1 4∈A 2 5∈A 3 9∈ /A Solution: 1 4 ∈ A: True 2 5 ∈ A: False 3 9∈ / A: True Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Ways of Representing Sets Sets can be represented in multiple ways depending on the context. In this lecture, we will discuss three common ways to represent sets: Roster or Tabular Form Set Builder Notation Venn Diagrams Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Roster or Tabular Form Definition: The roster or tabular form lists all elements of the set, separated by commas and enclosed in curly braces. Example: Let A be the set of natural numbers less than 5. A = {1, 2, 3, 4} Note: This form is useful for finite sets or sets where all elements are known. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Set Builder Notation Definition: Set builder notation describes the elements of a set by a property they satisfy, rather than listing them explicitly. Example: Let A be the set of all integers greater than 0 and less than 5. A = {x|x ∈ Z, 0 < x < 5} Note: This form is useful for describing large sets or sets with an easily defined property. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Venn Diagrams Definition: Venn Diagrams are used to represent sets visually. Each set is represented by a circle, and the relationships between sets (e.g., union, intersection) are represented by the overlap or separation of these circles. Example: Let A = {1, 2, 3} and B = {3, 4, 5}. In the Venn diagram, there is an overlap between A and B at element 3. Note: Venn Diagrams are particularly useful for representing set operations like union and intersection. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Venn Diagrams Figure: Venn diagram Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 1 Example: Let A be the set of all even numbers between 1 and 10. Represent A using: Roster Form; Set Builder Notation; and Venn Diagram. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 1 Example: Let A be the set of all even numbers between 1 and 10. Represent A using: Roster Form: A = {2, 4, 6, 8, 10} Set Builder Notation: A = {x|x is even and 1 < x ≤ 10} Venn Diagram: Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Classwork Question 1: Represent the set B = {x|x is a prime number less than 10} in roster form. Question 2: Represent the set C = {1, 2, 3, 4} using a Venn Diagram. Question 3: Write the set D, where D = {x|x ∈ N, x ≤ 7}, in roster form. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Take-Home Assignment Assignment 1: Represent the set E = {x|x is an odd number between 1 and 15} using roster form and set builder notation. Assignment 2: Create a Venn Diagram for two sets A and B, where A = {2, 4, 6, 8} and B = {4, 5, 6, 7}. Show their union and intersection. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Types of Sets Finite Sets: Sets with a limited number of elements. Example: A = {1, 2, 3, 4} Infinite Sets: Sets with an uncountable (unlimited) number of elements. Example: The set of all natural numbers N = {1, 2, 3,... } Empty Set (∅): A set with no elements. An empty set is denoted by ∅ or {} Singleton Set: A set with exactly one element. Example: D = {3} Universal Set: The set that contains all elements under consideration. The universal set is represented by the symbol U or E Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment What is a Subset? A set A is a subset of another set B, written A ⊆ B, if every element of A is also an element of B. Example: A = {1, 2}, B = {1, 2, 3, 4} In this case, A ⊆ B because all elements of A (i.e., 1 and 2) are in B. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Types of Subsets Proper Subset: A set A is a proper subset of B if A ⊆ B and A ̸= B. Denoted as A ⊂ B. Example: If A = {1, 2} and B = {1, 2, 3}, then A ⊂ B. Improper Subset: A set A is an improper subset of B if A = B or A is the empty set (∅). Example: A = B = {1, 2, 3}, then A ⊆ B (improper subset). Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example Problem: Given X = {a, b, c}, list all the subsets of X. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example Problem: Given X = {a, b, c}, list all the subsets of X. Solution: The subsets of X are: {}, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} There are 23 = 8 subsets. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Class Work Task: For the set A = {1, 2, 3, 4}, find: 1 All the subsets of A. 2 The proper subsets of A. 3 Verify that the number of subsets of A is 24. Hint: Use the formula 2n , where n is the number of elements in the set. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Assignment Homework: 1 List all the subsets of B = {x, y , z, w }. 2 How many proper subsets does the set B have? 3 For a set C with 5 elements, how many subsets and proper subsets does C have? Submission: Next class. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Definition of Superset Superset: A set A is said to be a superset of a set B if every element of B is also an element of A. Denoted as: A ⊇ B. If A ⊇ B, this implies that B ⊆ A, meaning B is a subset of A. If A ⊃ B and A ̸= B, then A is a proper superset of B. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example Example: Let A = {1, 2, 3, 4} and B = {2, 3}. Since every element of B is contained in A, we say that A is a superset of B. Thus, A ⊇ B. Verification: Check if all elements of B are in A: B = {2, 3} and A = {1, 2, 3, 4} Since 2 ∈ A and 3 ∈ A, it is true that B ⊆ A, or equivalently, A ⊇ B. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example Example: Let A = {a, b}, B = {a, b, c, d}, and C = {a, b}. Check whether the following statements are true: 1 A⊆B 2 A⊂B 3 B⊇C 4 A=C Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example Example: Let A = {a, b}, B = {a, b, c, d}, and C = {a, b}. Check whether the following statements are true: 1 A⊆B 2 A⊂B 3 B⊇C 4 A=C Solution: A ⊆ B is true since all elements of A are in B. A ⊂ B is true because A ̸= B but every element of A is in B. B ⊇ C is true because C ⊆ B. A = C is true because A and C have the same elements. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 2 Example: Let P = {x ∈ N : x ≤ 4} and Q = {1, 2, 3, 4, 5}. Determine whether P ⊂ Q and whether P ⊆ Q. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 2 Example: Let P = {x ∈ N : x ≤ 4} and Q = {1, 2, 3, 4, 5}. Determine whether P ⊂ Q and whether P ⊆ Q. Solution: P = {1, 2, 3, 4} and Q = {1, 2, 3, 4, 5}. P ⊂ Q is true because P ̸= Q, but all elements of P are in Q. P ⊆ Q is true because every element of P is in Q. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Classwork Solve the following: 1 Let A = {2, 4, 6} and B = {2, 4, 6, 8}. Determine whether A ⊂ B and B ⊇ A. 2 Let X = {a, b, c}, Y = {a, b, c, d}. Find if X ⊂ Y and Y ⊇ X. Discuss your answers with your classmate. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Take-Home Assignment Assignment: 1 Let S = {5, 10, 15, 20} and T = {10, 15, 20, 25}. Is S ⊆ T ? Why or why not? 2 Given A = {1, 2, 3, 4, 5} and B = {3, 4, 5}, find if A ⊆ B or B ⊆ A. 3 Find three real-life examples where proper subsets and supersets can be applied. Submit by next class. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Power Set Definition: The power set of a set A is the set of all possible subsets of A. Notation: P(A) Example: If A = {1, 2}, then P(A) = {∅, {1}, {2}, {1, 2}} Formula: For a set with n elements, the power set has 2n elements. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example Question: Find the power set of A = {a, b}. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example Question: Find the power set of A = {a, b}. Solution: P(A) = {∅, {a}, {b}, {a, b}} Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Classwork Given the set C = {x, y , z}, find its power set P(C ). Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Introduction to Operations on Sets Set operations help us work with and combine sets in meaningful ways. We will discuss key operations such as Union, Intersection, Difference, and Complement using both set notation and set builder notation. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Union of Sets Union: The union of two sets contains all elements from both sets without duplication. Notation: A ∪ B = {x : x ∈ A or x ∈ B} or A ∪ B = {x|x ∈ A or x ∈ B} Example: Let A = {1, 2, 3} and B = {3, 4, 5}. A ∪ B = {1, 2, 3, 4, 5} Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Intersection of Sets Intersection: The intersection of two sets contains only the elements common to both sets. Set Notation: A ∩ B = {x : x ∈ A and x ∈ B} or A ∩ B = {x|x ∈ A and x ∈ B} Example: Let A = {1, 2, 3} and B = {3, 4, 5}. A ∩ B = {3} Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Difference of Sets Difference: Set Notation: A − B = A \ B = {x : x ∈ A and x ∈ / B} or A − B = A \ B = {x|x ∈ A and x ∈ / B} The difference of two sets contains elements in A but not in B. Example: Let A = {1, 2, 3} and B = {3, 4, 5}. A − B = {1, 2} Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 1 Example: Let A = {a, b, c}, B = {b, c, d}, and U = {a, b, c, d, e}. Find A ∪ B, A ∩ B, A − B, and A′. Solution: A ∪ B = {a, b, c, d} A ∩ B = {b, c} A − B = {a} A′ = {d, e} Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Further Examples Example 1: Real-Life Application of Union Consider two groups of students: Group A: Students taking Mathematics A = {Alice, Bob, Charlie} Group B: Students taking Science B = {Charlie, David, Eva} The union A ∪ B includes all students taking either Mathematics or Science: A ∪ B = {Alice, Bob, Charlie, David, Eva} Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Further Examples Example 2: Real-Life Application of Intersection From the previous groups, the intersection A ∩ B consists of students enrolled in both subjects: Consider two groups of students: Group A: Students taking Mathematics A = {Alice, Bob, Charlie} Group B: Students taking Science B = {Charlie, David, Eva} A ∩ B = {Charlie} Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Set Complement The complement of a set A, denoted by A′ or A, or Ac consists of all elements in the universal set U that are not in A. Mathematically, A′ = U − A. Set Notation: Ac or A′ = {x : x ∈ U and x ∈ / A} OR A′ = {x|x ∈ U and x ∈ / A} Complements are useful in set operations and Venn diagrams. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Properties of the Complement Property 1: Double Complement (Involution Law) (A′ )′ = A The complement of the complement of set A is A itself. Property 2: Complement of Universal Set U′ = ∅ The complement of the universal set is the empty set. Property 3: Complement of the Empty Set ∅′ = U The complement of the empty set is the universal set. Property 4: De Morgan’s Laws (A ∪ B)′ = A′ ∩ B ′ , (A ∩ B)′ = A′ ∪ B ′ These laws describe how the complement of unions and intersections can be expressed in terms of the complements. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 1 Example: Let U = {1, 2, 3, 4, 5, 6, 7, 8} be the universal set, and A = {2, 4, 6}. Find the complement of set A. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 1 Example: Let U = {1, 2, 3, 4, 5, 6, 7, 8} be the universal set, and A = {2, 4, 6}. Find the complement of set A. Solution: A′ = U − A = {1, 3, 5, 7, 8} Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 2 Example: Let U = {a, b, c, d, e, f }, A = {a, b, c}, and B = {c, d, e}. Verify De Morgan’s law for the complement of the union of sets A and B. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 2 Example: Let U = {a, b, c, d, e, f }, A = {a, b, c}, and B = {c, d, e}. Verify De Morgan’s law for the complement of the union of sets A and B. Solution: A ∪ B = {a, b, c, d, e} (A ∪ B)′ = U − (A ∪ B) = {f } A′ = {d, e, f }, B ′ = {a, b, f } A′ ∩ B ′ = {f } Thus, (A ∪ B)′ = A′ ∩ B ′ , confirming De Morgan’s law. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Classwork Question 1: Let U = {10, 20, 30, 40, 50, 60} and C = {20, 40, 60}. Find C ′. Question 2: Given U = {x, y , z, w }, A = {x, y }, and B = {y , z}, verify De Morgan’s laws for A and B. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Take-Home Assignment Assignment 1: Let U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {2, 3, 5}, and B = {4, 6, 7}. Find (A ∪ B)′ and verify it using De Morgan’s law. Assignment 2: For the universal set U = {a, b, c, d, e, f }, and sets X = {a, b, c} and Y = {c, d, e}, show that (X ∩ Y )′ = X ′ ∪ Y ′ using Venn diagrams. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Take-Home Assignment Assignment 1: Let U = {a, b, c, d, e}, A = {a, b}, and B = {b, c, d}. Find A ∪ B, A ∩ B, A − B, and (A ∪ B)′. Assignment 2: Given U = {1, 2, 3, 4, 5, 6, 7, 8}, A = {1, 2, 3}, and B = {3, 4, 5}, verify De Morgan’s laws for the sets. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment What is Symmetric Difference? The symmetric difference of two sets A and B is the set of elements that are in either A or B, but not in both. It is denoted as A∆B = (A \ B) ∪ (B \ A). In simpler terms, it gives elements that are in A or B, but not in their intersection. Example: If A = {1, 2, 3} and B = {3, 4, 5}, then A∆B = {1, 2, 4, 5}. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Properties of Symmetric Difference Commutative Property: A∆B = B∆A. Associative Property: (A∆B)∆C = A∆(B∆C ). Identity: A∆∅ = A. Symmetric Difference of a Set with Itself: A∆A = ∅. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 1 Example 1: Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Find the symmetric difference A∆B. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 1 Example 1: Let A = {1, 2, 3, 4} and B = {3, 4, 5, 6}. Find the symmetric difference A∆B. Solution: A \ B = {1, 2} (elements in A but not in B). B \ A = {5, 6} (elements in B but not in A). Therefore, A∆B = {1, 2, 5, 6}. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 2 Example 2: Let A = {a, b, c} and B = {b, c, d, e}. Find A∆B. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Worked Example 2 Example 2: Let A = {a, b, c} and B = {b, c, d, e}. Find A∆B. Solution: A \ B = {a}. B \ A = {d, e}. Therefore, A∆B = {a, d, e}. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Classwork Classwork: Let A = {2, 4, 6, 8} and B = {4, 8, 12}. Find the symmetric difference A∆B. Instructions: Solve this problem in your notebooks and be ready to discuss the solution in class. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Take-Home Assignment Assignment: Let C = {1, 3, 5, 7, 9} and D = {3, 6, 9, 12}. Find: 1 C ∆D 2 D∆C 3 Verify the commutative property: Is C ∆D = D∆C ? Due Date: Submit the assignment in the next class. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Venn Diagrams Definition: Venn diagrams are used to represent sets and their relationships visually. Each set is represented by a circle. Example: Union: Two overlapping circles, where both circles and the intersection are shaded. Figure: Union of Sets A and B Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Venn Diagrams Example: Intersection: Two overlapping circles, with only the intersection shaded. Figure: Intersection of Sets A and B Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Venn Diagrams Example: Complement: A circle inside a rectangle (universal set), with the outside of the circle shaded. Figure: Complement of set A Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Applications of Set Theory Data organization and classification Probability and statistics Logic and mathematical reasoning Computer science and database management Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Summary Sets are foundational in mathematics. Understanding operations on sets is crucial for advanced topics. Venn diagrams are helpful tools for visualizing set relationships. Introduction to Subsets Types of Subsets Worked Example Worked Example Class Work Assignment Take-Home Assignment Create your own Venn diagram with two sets, and identify the union, intersection, and differences. Find the power set for the set D = {1, 2, 3, 4}.